Is it possible to use a frames marker name and append it to the file name upon export?
No, this isn't supported, however there are some options you could try instead.


Stamp the marker name into the file and read the marker name back out later. (JPEG, EXR, PNG support writing stamp data into the file) 
Write a pre-render handler to impliment this with a python script, so before rendering, change the name based on the marker (I didn't check this works, only some settings can be changed while rendering). (See: http://www.blender.org/documentation/blender_python_api_2_71_release/bpy.app.handlers.html)
Render marker ranges - See: https://blender.stackexchange.com/a/15373/55 triggered by a python script.